Advanced Topics in Data Mining: Association Rules

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Advanced Topics in Data MiningAssociation Rules
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What Is Association Mining?

• Association Rule Mining
• Finding frequent patterns, associations,
  correlations, or causal structures among item
  sets in transaction databases, relational
  databases, and other information repositories
• Applications
• Market basket analysis (marketing strategy items
  to put on sale at reduced prices),
  cross-marketing, catalog design, shelf space
  layout design, etc
• Examples
• Rule form Body Head Support, Confidence.
• buys(x, Computer) buys(x, Software) 2,
  60
• major(x, CS) takes(x, DB) grade(x, A)
  1, 75

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Market Basket Analysis
Typically, association rules are considered
interesting if they satisfy both a minimum
support threshold and a minimum confidence
threshold.
4
Rule Measures Support and Confidence

• Let minimum support 50, and minimum confidence
  50, we have
• A ? C 50, 66.6
• C ? A 50, 100

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Support Confidence
6
Association Rule Basic Concepts

• Given
• (1) database of transactions,
• (2) each transaction is a list of items
  (purchased by a customer in a visit)
• Find all rules that correlate the presence of one
  set of items with that of another set of items
• Find all the rules A ? B with minimum confidence
  and support
• support, s, P(A ? B)
• confidence, c, P(BA)

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Association Rule MiningA Road Map

• Boolean vs. quantitative associations (Based on
  the types of values handled in the rule set)
• buys(x, SQLServer) buys(x, DM Book) ?
  buys(x, DBMiner) 0.2, 60
• age(x, 30..39) income(x, 42..48K) ? buys(x,
  PC) 1, 75
• Single dimension vs. multiple dimensional
  associations
• Single level vs. multiple-level analysis (Based
  on the levels of abstractions involved in the
  rule set)

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Terminologies

• Item
• I1, I2, I3,
• A, B, C,
• Itemset
• I1, I1, I7, I2, I3, I5,
• A, A, G, B, C, E,
• 1-Itemset
• I1, I2, A,
• 2-Itemset
• I1, I7, I3, I5, A, G,

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Terminologies

• K-Itemset
• If the length of the itemset is K
• Frequent K-Itemset
• If the length of the itemset is K and the itemset
  satisfies a minimum support threshold.
• Association Rule
• If a rule satisfies both a minimum support
  threshold and a minimum confidence threshold

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Analysis

• The number of itemsets of a given cardinality
  tends to grow exponentially

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Mining Association Rules Apriori Principle
Min. support 50 Min. confidence 50

• For rule A ? C
• support support(A ? C) 50
• confidence support(A ? C)/support(A)
  66.6
• The Apriori principle
• Any subset of a frequent itemset must be frequent

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Mining Frequent Itemsets the Key Step

• Find the frequent itemsets the sets of items
  that have minimum support
• A subset of a frequent itemset must also be a
  frequent itemset
• i.e., if AB is a frequent itemset, both A and
  B should be a frequent itemset
• Iteratively find frequent itemsets with
  cardinality from 1 to k (k-itemset)
• Use the frequent itemsets to generate
  association rules

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Example
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Apriori Algorithm
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Apriori Algorithm
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Apriori Algorithm
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Example of Generating Candidates

• L3abc, abd, acd, ace, bcd
• Self-joining L3L3
• abcd from abc and abd
• acde from acd and ace
• Pruning
• acde is removed because ade is not in L3
• C4abcd

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Another Example 1
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Another Example 2
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Is Apriori Fast Enough? Performance Bottlenecks

• The core of the Apriori algorithm
• Use frequent (k1)-itemsets to generate candidate
  frequent k-itemsets
• Use database scan to collect counts for the
  candidate itemsets
• The bottleneck of Apriori
• Huge candidate sets
• 104 frequent 1-itemset will generate 107
  candidate 2-itemsets
• To discover a frequent pattern of size 100, e.g.,
  a1, a2, , a100, one needs to generate 2100 ?
  1030 candidates.
• Multiple scans of database
• Needs (n 1) scans, n is the length of the
  longest pattern

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Demo-IBM Intelligent Minner
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Demo Database
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Methods to Improve Aprioris Efficiency

• Hash-based itemset counting A k-itemset whose
  corresponding hashing bucket count is below the
  threshold cannot be frequent
• Transaction reduction A transaction that does
  not contain any frequent k-itemset is useless in
  subsequent scans
• Partitioning Any itemset that is potentially
  frequent in DB must be frequent in at least one
  of the partitions of DB
• Sampling mining on a subset of given data, lower
  support threshold a method to determine the
  completeness

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Partitioning
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Hash-Based Itemset Counting



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Compare Apriori DHP (Direct Hash Pruning)
Apriori
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Compare Apriori DHP
DHP
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DHP Database Trimming
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DHP (Direct Hash Pruning)

• A database has four transactions
• Let min_sup 50

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Example Apriori
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Example DHP
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Example DHP
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Example DHP
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Mining Frequent Patterns Without Candidate
Generation

• Compress a large database into a compact,
  Frequent-Pattern tree (FP-tree) structure
• highly condensed, but complete for frequent
  pattern mining
• avoid costly database scans
• Develop an efficient, FP-tree-based frequent
  pattern mining method
• A divide-and-conquer methodology decompose
  mining tasks into smaller ones
• Avoid candidate generation sub-database test
  only!

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Construct FP-tree from a Transaction DB
39
Construction Steps

• Scan DB once, find frequent 1-itemset (single
  item pattern)
• Order frequent items in frequency descending
  order
• Sorting DB according to the frequency descending
  order
• Scan DB again, construct FP-tree

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Benefits of the FP-Tree Structure

• Completeness
• never breaks a long pattern of any transaction
• preserves complete information for frequent
  pattern mining
• Compactness
• reduce irrelevant informationinfrequent items
  are gone
• frequency descending ordering more frequent
  items are more likely to be shared
• never be larger than the original database (if
  not count node-links and counts)
• Compression ratio could be over 100

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Frequent Pattern Growth
Order frequent items in frequency descending order

• For I5
• I1, I5, I2, I5, I2, I1, I5
• For I4
• I2, I4

• For I3
• I1, I3, I2, I3, I2, I1, I3
• For I1
• I2, I1

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Frequent Pattern Growth

• For I5
• I1, I5, I2, I5, I2, I1, I5
• For I4
• I2, I4
• For I3
• I1, I3, I2, I3, I2, I1, I3
• For I1
• I2, I1

Sub DB
Trimming Databases
Sub DB
Sub DB
FP-tree
Sub DB
Conditional FP-tree from Conditional Pattern-Base
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Conditional FP-tree
Conditional FP-tree from Conditional
Pattern-Base for I3
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Mining Results Using FP-tree

• For I5 (???NULL)
• Conditional Pattern Base
• (I2I11), (I2I1I31)
• Conditional FP-tree

• Generate Frequent Itemsets
• I22 Rule I2I52
• I12 Rule I1I52
• I2I12 Rule I2I1I52

? NULL 2
Item ID Support count Node Link
I2 2
I1 2
? I2 2
? I1 2
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Mining Results Using FP-tree

• For I4
• Conditional Pattern Base
• (I2I11), (I21)
• Conditional FP-tree

• Generate Frequent Itemsets
• I22 Rule I2I42

? NULL 2
Item ID Support count Node Link
I2 2
? I2 2
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Mining Results Using FP-tree

• For I3
• Conditional Pattern Base
• (I2I12), (I22), (I12)
• Conditional FP-tree

? NULL 4
Item ID Support count Node Link
I2 4
I1 4
? I2 4
? I1 2
? I1 2
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Mining Results Using FP-tree

• For I1/I3
• Conditional Pattern Base
• (NULL2), (I22)
• Conditional FP-tree

• Generate Frequent Itemsets
• Null4 Rule I1I34
• I22 Rule I2I1I32

? NULL 4
Item ID Support count Node Link
I2 2
? I2 2
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Mining Results Using FP-tree

• For I2/I3
• Conditional Pattern Base
• (NULL4)
• Conditional FP-tree

• Generate Frequent Itemsets
• Null Rule I2I34

? NULL 4
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Mining Results Using FP-tree

• For I1
• Conditional Pattern Base
• (NULL2), (I24)
• Conditional FP-tree

• Generate Frequent Itemsets
• I24 Rule I2I14

? NULL 6
Item ID Support count Node Link
I2 4
? I2 4
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Mining Results Using FP-tree
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Mining Frequent PatternsUsing FP-tree

• General idea (divide-and-conquer)
• Recursively grow frequent pattern path using the
  FP-tree
• Method
• For each item, construct its conditional
  pattern-base, and then its conditional FP-tree
• Repeat the process on each newly created
  conditional FP-tree
• Until the resulting FP-tree is empty, or it
  contains only one path (single path will generate
  all the combinations of its sub-paths, each of
  which is a frequent pattern)

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Major Steps to Mine FP-tree

• Construct conditional pattern base for each item
  in the FP-tree
• Construct conditional FP-tree from each
  conditional pattern-base
• Recursively mine conditional FP-trees and grow
  frequent patterns obtained so far
• If the conditional FP-tree contains a single
  path, simply enumerate all the patterns

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Virtual Items in Association Mining

• Different region exhibit different selling
  patterns. Thus, including as virtual item the
  information on the location or the type of stores
  (existing or new) where the purchase was made
  will enable the comparisons between locations or
  types within a single chain
• Virtual item may include information on whether
  the purchase was made with cash, a credit card or
  check. The inclusion of such virtual item allows
  to analyze the association between the payment
  method and items purchased.
• Virtual item may include information on the day
  of the week or the time of the day the
  transaction occurred. The inclusion of such
  virtual item allows to analyze the association
  between the transaction time and items purchased

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Virtual Items An Example
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Dissociation Rules

• A dissociation rule is similar to an association
  rule except that it can have not item-name in
  the condition or the result of the rule
• A and not B C    
• A and D not E
• Dissociation rules can be generated by a simple
  adaptation of the association rule analysis

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Discussions

• The size of a typical transaction grows because
  it now includes inverted items
• The total number of items used in the analysis
  doubles
• Since the amount of computation grows
  exponentially with the number of items, doubling
  the number of items seriously degrades
  performance
• The frequency of the inverted items tends to be
  much larger than the frequency of the original
  items. So, it tends to produce rules in which all
  items are inverted. These rules are less likely
  to be actionable.
• not A and not B not C
• It is useful to invert only the most frequent
  items in the set used for analysis. It is also
  useful to invert some items whose inverses are of
  interest.

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Interestingness Measurements

• Subjective Measures
• A rule (pattern) is interesting if
• it is unexpected (surprising to the user)
• actionable (the user can do something with it)
• Objective Measures
• Two popular measurements
• Support
• confidence

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Criticism to Support and Confidence

• Example 1
• Among 5000 students
• 3000 play basketball
• 3750 eat cereal
• 2000 both play basket ball and eat cereal
• play basketball ? eat cereal 40, 66.7 is
  misleading because the overall percentage of
  students eating cereal is 75 which is higher
  than 66.7.

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Criticism to Support and Confidence

• Example 2
• X and Y positively correlated,
• X and Z, negatively related
• support and confidence of
• XgtZ dominates
• We need a measure of dependent or correlated
  events

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Criticism to Support and Confidence

• Improvement (Correlation)
• Taking both P(A) and P(B) in consideration
• P(AB)P(B)P(A), if A and B are independent
  events
• A and B negatively correlated, if the value is
  less than 1otherwise A and B positively
  correlated
• When improvement is less than 1, negating the
  result produces a better rule
• X gt NOT Z

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Multiple-Level Association Rules

• Items often form hierarchy
• Items at the lower level are expected to have
  lower support
• Rules regarding itemsets at appropriate levels
  could be quite useful
• Transaction database can be encoded based on
  dimensions and levels
• We can explore multi-level mining

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Transaction Database
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Concept Hierarchy
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Mining Multi-Level Associations

• A top_down, progressive deepening approach
• First find high-level strong rulesmilk bread
  20, 60
• Then find their lower-level weaker rules2
  milk wheat bread 6, 50
• Variations at mining multiple-level association
  rules.
• Cross-level association rules (Generalized Asso.
  Rules)2 milk Wonder wheat bread
• Association rules with multiple, alternative
  hierarchies2 milk Wonder bread

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Multi-level Association Uniform Support vs.
Reduced Support

• Uniform Support the same minimum support for all
  levels
• One minimum support threshold is needed.
• Lower level items do not occur as frequently. If
  support threshold
• too high ? miss low level associations
• too low ? generate too many high level
  associations
• Reduced Support reduced minimum support at lower
  levels
• There are 4 search strategies
• Level-by-level independent
• Level-cross filtering by k-itemset
• Level-cross filtering by single item
• Controlled level-cross filtering by single item

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Uniform Support
?

• Optimization Technique
• The search avoids examining itemsets containing
  any item whose ancestors do not have minimum
  support.

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Uniform Support An Example
L1 2,3,4,5,6 L2 23,24,25,26,34,45,46,56 L3
234,245,246,256,456 L4 2456
min_sup50
L1 2,3,4 L2 23,24,34 L3 234
min_sup60
L1 2,3,6,8,9 L2 23,68,69,89 L3 689
min_sup50
L1 9
min_sup60
68
Uniform Support An Example
All
Cheese
Crab
Milk
3
1
2
1
2
6
3
5
Kings Crab
Sunset Milk
Dairyland Milk
Dairyland Cheese
Best Cheese
Bread
Apple
Pie
4
5
6
4
9
7
10
8
Best Bread
Wonder Bread
Goldenfarm Apple
Westcoast Bread
Tasty Pie
69
Uniform Support An Example
L1 2,3,4,5,6 L2 23,24,25,26,34,45,46,56 L3
234,245,246,256,456 L4 2456
min_sup50
Apriori/DHP FP Growth
(1)
(2)
min_sup50
L1 2,3,6,8,9
Scan DB
C2 23,36,29,69,28,68,39,89 C3 239,369,289,689
(3)
min_sup50
Scan DB
L2 23,68,69,89 L3 689
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Uniform Support An Example
All
Cheese
Crab
Milk
3
1
2
1
2
6
3
5
Kings Crab
Sunset Milk
Dairyland Milk
Dairyland Cheese
Best Cheese
Bread
Apple
Pie
4
5
6
4
9
7
10
8
Best Bread
Wonder Bread
Goldenfarm Apple
Westcoast Bread
Tasty Pie
71
Reduced Support

• Each level of abstraction has its own minimum
  support threshold.

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Search Strategies forReduced Support

• There are 4 search strategies
• Level-by-level independent
• Full-Breadth Search
• No pruning
• No background knowledge of frequent itemsets is
  used for pruning
• Level-cross filtering by single item
• An item at the ith level is examined if and only
  if its parent node at the (i-1)th level is
  frequent.
• Level-cross filtering by k-itemset
• An k-itemset at the ith level is examined if and
  only if its corresponding parent k-itemset at the
  (i-1)th level is frequent.
• Controlled level-cross filtering by single item

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Level-Cross Filtering by Single Item
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Reduced Support An Example
L1 2,3,4 L2 23,24,34 L3 234
min_sup60
Apriori/DHP FP Growth
(1)
(1)
min_sup50
L1 2,3,6,9
Scan DB
(2)
min_sup50
L2 23,69
Apriori/DHP FP Growth
(3)
75
Reduced Support An Example
All
Cheese
Crab
Milk
3
1
2
1
2
6
3
5
Kings Crab
Sunset Milk
Dairyland Milk
Dairyland Cheese
Best Cheese
Bread
Apple
Pie
4
5
6
4
9
7
10
8
Best Bread
Wonder Bread
Goldenfarm Apple
Westcoast Bread
Tasty Pie
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Level-Cross Filtering by K-Itemset
?
?
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Reduced Support An Example
L1 2,3,4 L2 23,24,34 L3 234
min_sup60
Apriori/DHP FP Growth
(1)
(2)
min_sup50
L1 2,3,6,9
Scan DB
C2 23,36,29,69,39 C3 239,369
(3)
min_sup50
Scan DB
L2 23,69
78
Reduced Support An Example
All
Cheese
Crab
Milk
3
1
2
1
2
6
3
5
Kings Crab
Sunset Milk
Dairyland Milk
Dairyland Cheese
Best Cheese
Bread
Apple
Pie
4
5
6
4
9
7
10
8
Best Bread
Wonder Bread
Goldenfarm Apple
Westcoast Bread
Tasty Pie
79
Reduced Support

• Level-by-level independent
• It is very relaxed in that it may lead to
  examining numerous infrequent items at low
  levels, finding associations between items of
  little importance.
• Level-cross filtering by k-itemset
• It allows the mining system to examine only the
  children of frequent k-itemsets.
• This restriction is very strong in that there
  usually are not many k-itemsets.
• Many valuable patterns may be filtered out.
• Level-cross filtering by single item
• A compromise between the above two approaches
• This method may miss associations between low
  level items that are frequent based on a reduced
  minimum support, but whose ancestors do not
  satisfy minimum support.

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Controlled Level-Cross Filtering by Single Item
81
Reduced Support An Example
L1 2,3,4 L2 23,24,34 L3 234
min_sup60
Apriori/DHP FP Growth
level_passage_sup50 L1 2,3,4,5,6
(1)
(1)
min_sup50
L1 2,3,6,8,9
Scan DB
(2)
L2 23,68,69,89 L3 689
min_sup50
Apriori/DHP FP Growth
(3)
82
Reduced Support An Example
All
Cheese
Crab
Milk
3
1
2
1
2
6
3
5
Kings Crab
Sunset Milk
Dairyland Milk
Dairyland Cheese
Best Cheese
Bread
Apple
Pie
4
5
6
4
9
7
10
8
Best Bread
Wonder Bread
Goldenfarm Apple
Westcoast Bread
Tasty Pie
83
Multi-Dimensional Association

• Single-Dimensional (Intra-Dimension) Rules
  Single Dimension (Predicate) with Multiple
  Occurrences.
• buys(X, milk) ? buys(X, bread)
• Multi-Dimensional Rules ? 2 Dimensions
• Inter-dimension association rules (no repeated
  predicates)
• age(X,19-25) ? occupation(X,student) ?
  buys(X,coke)
• hybrid-dimension association rules (repeated
  predicates)
• age(X,19-25) ? buys(X, popcorn) ? buys(X,
  coke)

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Summary

• Association rule mining
• Probably the most significant contribution from
  the database community in KDD
• A large number of papers have been published
• Some Important Issues
• Generalized Association Rules
• Multiple-Level Association Rules
• Association Analysis in Other Types of Data
• Spatial Data, Multimedia Data, Time Series Data,
  etc.
• Weighted Association Rules
• Quantitative Association Rules

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Weighted Association Rules

• Why Weighted Association Analysis?
• In previous work, all items in a transactional
  database are treated uniformly
• Items are given weights to reflect their
  importance to the user
• The weights may correspond to special promotions
  on some products, or the profitability of
  different items
• Some products may be under promotion and hence
  are more interesting, or some products are more
  profitable and hence rules concerning them are of
  greater values

86
Weighted Association Rules

• A simple attempt to solve this problem is to
  eliminate the items with small weights
• However, a rule for a heavy weighted item may
  also consist of low weighted items
• Is Apriori algorithm feasible?
• Apriori algorithm depends on the downward closure
  property which governs that subsets of a frequent
  itemset are also frequent
• However, it is not true for the weighted case

87
Weighted Association RulesAn Example

• Total Benefits 500
• Benefits for the First Transaction
  (403030202010 10)160
• Benefits for the Second Transaction (403020
  201010)130
• Benefits for the Third Transaction
  (4030201010) 110
• Benefits for the Fourth Transaction
  (3030201010)100
• Suppose Weighted_Min_Sup 40
• Minimum Benefits 500 40 200

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An Example

• Minimum Benefits 500 40 200
• Itemset 3,5,6,7
• Benefits 70
• Support Count (Frequency) 3
• 70 3 210 gt 200 ?3,5,6,7is a Frequent
  Itemset
• Itemset 3,5,6
• Benefits 60
• Support Count ( Frequency) 3
• 60 3 180 lt 200 ?3,5,6 is not a Frequent
  Itemset

Apriori Principle can not be applied!
89
K-Support Bound

• If Y is a frequent q-itemset
• Support_Count(Y) ? (Weighted_Min_Sup
  Total_Benefits) / Benefits(Y)
• Example
• 3,5,6,7 is a Frequent 4-Itemset
• Support_Count(3,5,6,7) 3 ? (40 500) /
  Benefits(3,5,6,7) (40 500) / 70 2.857
• If X is a frequent k-itemset containing q-itemset
  Y
• Minimum_Support_Count(X) ? (Weighted_Min_Sup
  Total_Benefits) / (Benefits(Y) (k-q) Maximum
  Remaining Weights)
• Example
• X is a Frequent 5-Itemset containing 3,5,6,7
• Minimum_Support_Count(X) ? (40 500) / (70
  40) 1.81

K-Support Bound
90
K-Support Bound

• Itemset 1,2
• Benefits 70
• Support_Count(1,2) 1 lt (40 500) /
  Benefits(1,2) (40 500) / 70 2.857
• 1,2 is not a Frequent Itemset
• If X is a frequent 3-itemset containing 1,2
• Minimum_Support_Count(X) ? (40 500) / (70
  30) 2
• But, Maximum_Support_Count(X) 1
• No frequent 3-itemsets containing 1,2
• If X is a frequent 4-itemset containing 1,2
• Minimum_Support_Count(X) ? (40 500) / (70 30
  20) 1.667
• But, Maximum_Support_Count(X) 1
• No frequent 4-itemsets containing 1,2
• Similarly, no frequent 5, 6, 7-itemsets
  containing 1,2
• The algorithm is designed based on this k-support
  bound

91
MINWAL Algorithm
92
Step by Step

• Input Product Transactional Databases

Weighted_Min_Sup 50
Total Profits 1380
93
Step 2, 7

• Search(D)
• This subroutine finds out the maximum transaction
  size in that transactional database D
• Size 4 in this case
• Counting(D, w)
• This subroutine cumulates the support counts of
  the 1-itemsets
• The k-support bounds of each 1-itemset will be
  calculated, and the 1-itemsets with support
  counts greater than any of the k-support bounds
  will be kept in C1

94
Step 7
?
K-Support Bound ? (50 1380) / (10 90) 6.9
95
Step 11

• Join(Ck-1)
• The Join step generates Ck from Ck-1 as Apriori
  Algorithm
• If we have 1, 2, 3, 1, 2, 4 in Ck-1 1, 2, 3,
  4 will be generated in Ck
• In this case,
• C1 1 (4), 2 (5), 4 (6), 5 (7)
• C2 Join(C1) 12, 14, 15, 24, 25, 45

Support_Count
96
Step 12

• Prune(Ck)
• The itemset will be pruned in either of the
  following cases
• A subset of the candidate itemset in Ck does not
  exist in Ck-1
• Estimate an upper bound on the support count (SC)
  of the joined itemset X, which is the minimum
  support count among the k different (k-1)-subsets
  of X in Ck-1. If the estimated upper bound on
  the support count shows that the itemset X cannot
  be a subset of any large itemset in the coming
  passes (from the calculation of k-support bounds
  for all itemsets), that itemset will be pruned
• In this case,
• C2 Prune(C2) 12 (4), 14 (4), 15 (4), 24 (5),
  25 (5), 45 (6)
• Using K-Support-Bound

Estimated_Support_Count
97
Step 12

• Prune(Ck)
• Using K-Support-Bound (No one is pruned)

98
Step 13

• Checking(Ck, D)
• Scan DB Generate Lk

?
?
C2
L2
99
Step 11, 12

• Join(C2)
• C2 15 (4), 24 (5), 25 (5), 45 (6)
• C3 Join(C2) 245
• Prune(C3)
• C3 Prune(C3) 245 (5)
• Using K-Support-Bound (No one is pruned)

100
Step 13

• Checking(C3, D) Scan DB
• C3 L3 245
• Finally, L 45, 245

101
Step 15

• Generate Rules for L 45, 245
• 4 ? 5 (confidence 100)
• 5 ? 4 (confidence 85.7)
• 24 ? 5 (confidence 100)
• 25 ? 4 (confidence 100)
• 45 ? 2 (confidence 83.3)
• 2 ? 45 (confidence 100)
• 4 ? 25 (confidence 83.3)
• 5 ? 24 (confidence 71.4)

Min_Conf90
102
Generalized Association Rules
103
Quantitative Association Rules

• Let min_sup 50, we have
• A, B 60
• B, D 70
• A, B, D 50

• A(1..2), B(3) 50
• A(1..2), B(3..5) 60
• A(1..2), B(1..5) 60
• B(1..5), D(1..2) 60
• B(3..5), D(1..3) 60
• B(1..5), D(1..3) 70
• B(1..3), D(1..2) 50
• B(3..5), D(1..2) 50
• A(1..2), B(3..5), D(1..2) 50

?